Kronecker sums to construct Hadamard full propelinear codes of type CnQ₈

Publicació amb motiu de la 21st Conference on Applications of Computer Algebra (July 20-24, 2015, Kalamata, Greece

Ranks and kernels of codes from generalized Hadamard matrices

The ranks and kernels of generalized Hadamard matrices are studied. It is proved that any generalized Hadamard matrix H(q, λ) over Fq , q > 3, or q = 3 and gcd(3, λ) ≠ 1, generates a self-orthogonal code. This result puts a natural upper bound on the rank of the generalized Hadamard matrices. Lower and upper bounds are given for the dimension of the kernel of the corresponding generalized Hadamard codes. For specific ranks and dimensions of the kernel within these bounds, generalized Hadamard codes are constructed

On Degree-d Zero-Sum Sets of Full Rank

A set S⊆Fn2 is called degree-d zero-sum if the sum ∑s∈Sf(s) vanishes for all n-bit Boolean functions of algebraic degree at most d. Those sets correspond to the supports of the n-bit Boolean functions of degree at most n−d−1. We prove some results on the existence of degree-d zero-sum sets of full rank, i.e., those that contain n linearly independent elements, and show relations to degree-1 annihilator spaces of Boolean functions and semi-orthogonal matrices. We are particularly interested in the smallest of such sets and prove bounds on the minimum number of elements in a degree-d zero-sum set of rank n. The motivation for studying those objects comes from the fact that degree-d zero-sum sets of full rank can be used to build linear mappings that preserve special kinds of nonlinear invariants, similar to those obtained from orthogonal matrices and exploited by Todo, Leander and Sasaki for breaking the block ciphers Midori, Scream and iScream

Invariants de matrius Hadamard en MAGMA

L'objectiu d'aquest projecte ha estat generalitzar i integrar la funcionalitat de dos projectes anteriors que ampliaven el tractament que oferia el Magma respecte a les matrius de Hadamard. Hem implementat funcions genèriques que permeten construir noves matrius Hadamard de qualsevol mida per a cada rang i dimensió de nucli, i així ampliar la seva base de dades. També hem optimitzat la funció que calcula el nucli, i hem desenvolupat funcions que calculen la invariant Symmetric Hamming Distance Enumerator (SH-DE) proposada per Kai-Tai Fang i Gennian Gei que és més sensible per a la detecció de la no equivalència de les matrius Hadamard.El objetivo de este proyecto ha sido generalizar e integrar la funcionalidad de dos proyectos anteriores que ampliaban el tratamiento que ofrecía el Magma respecto a las matrices Hadamard. Hemos implementado funciones genéricas que permiten construir nuevas matrices Hadamard de cualquier orden para cada rango y dimensión de núcleo, y así ampliar su base de datos. También hemos optimizado la función que calcula el núcleo, y hemos desarrollado funciones que calculan el invariante Symmetric Hamming Distance Enumerator (SHDE) propuesta por Kai-Ta i Fangy Gennian Ge que es más sensible en la detección de la no equivalencia de las matrices Hadamard.The aim of this project has been to generalize and to integrate the former two projects' functionality which extended Magma's treatement in relation to Hadamard matrices. We have implemented generic functions that allow us to construct new Hadamard matrices of any order for each possible pair of rank and dimension of kernel, and thus to extend its database. We have also optimized the function that computes the kernel, and we have developed functions that compute the Symmetric Hamming Distance Enumerator invariant proposed by Kai-Tai Fang and Gennian Ge, which is more sensitive for detecting the inequivalence of Hadamard matrices.Nota: Aquest document conté originàriament altre material i/o programari només consultable a la Biblioteca de Ciència i Tecnologia